Unit 8 Lesson 9.4ASpecial Right Triangles

CCSSG-SRT 8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.G-SRT 8.1: Derive and use the trigonometric ratios for special right triangles (30°,60°,90°and 45°,45°,90°).

Lesson Goals• Define a special right triangle• Identify the patterns in special

right triangles.• Use the special right triangle

theorems to find missing side lengths in right triangles.

ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers

Why are special right triangle theorems useful if we have the Pythagorean Theorem?

DefinitionSpecial Right Triangles

A right triangle with acute angles of 45o and 45o.

A right triangle with acute angles of 30o and 60o.

A

BC

45o

45o

D

FE

60o

30o

not in notes

You Try (not in notes)Find the missing angle measures.

7

7

xo

90x x

2 90x

45x

xo

You TryFind the missing side length. Leave answer in radical form.

2

2 c

2 2 22 2c 2 8c

8c

2 2c 45o

45o

You Try

7

7 c

Find c. Leave answer in radical form.

45o

45o

2

2

7

72 2 7 2

45o

45o

45o

45o

10

10 2

45o

45o

58 2

45o

45o

the hypotenuse is times as long as each leg.2

45o - 45o - 90o Triangle Theorem

2ss

s

A

BC

45o

45o

In a 45 -45 -90 triangle,

You Try

260o

30o

b 4

2 2 24 2 b 216 4 b

212 b

12b

2 3b

Find the missing side length. Leave answer in radical form.

You Try

760o

30o

7 3 c

22 27 7 3c 2 196c

196c

14c

Find the missing side length. Leave answer in radical form.

You Try

a60o

30o

21 3 42

22 242 21 3a

21764 1323a 2441 a

441a

21a

Find the missing side length. Leave answer in radical form.

260o

30o

2 3 4

760o

30o

7 3

14

2160o

30o

21 3

42

360o

30o

3 3 6

60o

30o

4 3 8

4

2n

the length of the hypotenuse is 2 the shorter leg

3n

D

FE

60o

30o

n

30o - 60o - 90o Triangle Theorem

In a 30°-60°-90° triangle,

and the longer leg is the shorter leg.3

Steps to help in solvingStep 1: Place the pattern on the triangle (make sure you use a different variable than the ones on the triangle)

Step 2: Setup 3 different equations based on what you have on each side of the triangle

Step 3: Solve the equation that only has one variable

Step 4: Use the answer from step 3 to solve other two sides.

Please note not all answers will be “pretty”

ExampleFind the value of x.

ExampleFind the value of x.

x

5 5

45o

5 2x

Example

12 x

45o

12 2x

12

2x

6 2 x

Find the value of x.

Example60o

30o

f

g

  4 3g 4  2 4f

8

Find the value of f and g.

10 3

3

Example

5 3m

5

3m

5 3

3m

5 32

3

60o

30o

n

5

m

  2n m

Find the value of m and n.

Summary

Why are special right triangle theorems useful if we have the Pythagorean Theorem?

Today’s Assignment

p. 554: 1 – 14, 43, 45, 47 – 49